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group schemes. For scheme $S$, $S$-subgroup $H\hookrightarrow G$ we want a short exact sequence + \[ 0 \to H(T) \to G(T) \to (G/H)(T) \to 0 \,.\] + But the presheaf $G/H$ is not generally a sheaf. + + As an ansatz, consider the yoneda embedding + \[ y:\operatorname{Sch}_S \hookrightarrow \operatorname{PSh}(\operatorname{Sch}_S) \,.\] + Grothendieck showed: the fpqc-topology is subcanonical, ie. presentable presheaves are sheaves in the fpqc-topology. Therefore, it may be useful to consider the fppf-sheafification of $(T\mapsto G(T)/H(T)).$ (why fppf and not fpqc: later) + + \begin{bem}Let be: + + LRS = category of locally ringed spaces + + Sch = category of schemes + + Aff = category of affine schemes + + All of these are full subcats of each other. + + For + \[ y:\operatorname{Sch}_S \hookrightarrow \operatorname{PSh}(\operatorname{Sch}_S) \,,\] + our goal is to consider the essential image of $y$ without reference to $\operatorname{Sch}_S$ + \end{bem} + +\begin{bem}[Ansatz] + Schemes are build from affine schemes through glueing on open immersions, i.e. every scheme is a colimit (coequalizer) of affine schemes and open immersions + \[ \coprod_{i,j\in I\;k\in J}\Spec(B_{i,j}) \,\substack{\to\\[-1em]\to}\, \coprod_{i\in I}\Spec (A_i) \to X \,.\] +\end{bem} + +\begin{bsp} + For scheme $X$ and open subschemes $U,V\subseteq X$ with $U\cup V=X$ the canonical map + \[ y(U)\amalg_{y(W)}y(V) \to y(X) \] + is not generally an isomorphism in $\operatorname{PSh}(\operatorname{Sch})$ for $W=U\cap V$. + + But after sheafification $a:\operatorname{PSh}(\operatorname{Sch})\to\operatorname{Sh}(\operatorname{Sch})$ we get two isomorphisms (since presheaves are reflective subcat of sheaves) + \[ a(y(U)\amalg_{y(W)}y(V)) \to a(y(U))\amalg_{a(y(W))}a(y(V)) \to a(y(X))=y(X) \,. \] +\end{bsp} + +Question: What is an open immersion between affine schemes? + +\begin{satz} + Let $f:X\to Y$ be a morphism of schemes. TFAE (?) + \begin{enumerate}[(i)] + \item $f$ is open immersion, + \item $f$ is étale, + \item $f$ is flat mono of locally finite presentation. + \end{enumerate} +\end{satz} + +\begin{bem} + A family $\{\Spec(A_i)\to\Spec(A)\}_{i\in I}$ of open immersions is an {open covering} iff it can be verfeinert by an open covering of the form $\{\Spec(A_{f_\alpha})\to\Spec(A)\}_{\alpha}$. + TFAE + \begin{enumerate} + \item $\{\Spec(A_{f_\alpha})\to\Spec(A)\}_{\alpha}$ is an open covering, + \item $A\to \prod_\alpha A_{f_\alpha}$ is faithfully flat, + \item $(1)=(f_\alpha\mid\alpha)\subseteq A$. + \end{enumerate} +\end{bem} + +$\Sh(\Sch) \simeq \Sh(\Aff)$ and there we only need open immersions: + +\begin{bem} + Following the preceding remark, we have the sites $\operatorname{Aff}^{\operatorname{Zar}}$ of affine schmes with zariksi topology. + + The restriction $\operatorname{PSh}(\operatorname{Sch})\to\operatorname{PSh}(\operatorname{Aff})$ induces an equivalence of cats $\operatorname{Sh}(\operatorname{Sch})\to\operatorname{Sh}(\operatorname{Aff})$ by the comparison lemma. + A quasi inverse is given by + \[ F\mapsto \hat F :X\mapsto \lim_{\Spec A\to X} F(\Spec A)\,.\] + The essential image of + \[ \Sch \xhookrightarrow{y} \Sh_{\Zar}(\Sch) \xrightarrow{i^\ast} \Sh_{\Zar}(\Aff) \] + consists exactly of those sheaves that can be written as coequalizer of a diagram of the form + \[ \coprod_{i,j,k}\Spec(A_{ijk})\,\substack{\to\\[-1em]\to}\,\coprod_i\Spec A_i \] +\end{bem} + +The problem is to check whether something is or is not a sheaf in $\Sh_{\Zar}(\Aff)$. + + +\begin{bem} + Let $\Psh$ denote $\Psh(\Aff)=\operatorname{Fun}(\operatorname{CRing},\operatorname{Set})$ and $\Aff=\operatorname{im}(y:\operatorname{CRing}^{\operatorname{op}}\to \operatorname{PSh})$. Set $S(R):=y(R)$ as an "affine scheme". + + For $A$ ring, $X\in\Psh$, $p\in X(A)$ we have + \[ p^\#:S(A)\to X \] + in $\Psh$ via $S(A)(R)\to X(R), \phi\mapsto X(\phi)(R)$. + + A subfunctor $U\hookrightarrow X$ in $\Psh$ is an equivalence class of monos in $\Psh$. +\end{bem} + +When is a subfunctor an open immersion? "Open immersion" is local on target, so we can check this on open coverings by affine schemes. These are of the shortly following form + +\begin{bem} + For an ideal $I\subseteq A$ we have decompostion $V(I)\to \Spec (A) \leftarrow D(I)$. For ringmorphism $\phi:A\to R$ have + \[ \Spec(\phi)^{-1}(D(I)=D(\phi(I)\cdot R)) \,. \] + Therefore + \[\begin{tikzcd} + \Spec(R) \ar[r] \ar[dr,dashed,"\exists !"] & \Spec(A) \ar[d] \\ + & D(I) + \end{tikzcd}\] + factorizes iff $\phi(I)\cdot R=R$. +\end{bem} + +\begin{definition} + \begin{itemize} + \item For ideal $I\subseteq A$ define subfunctor $S(A)_I\subseteq S(A)$ via + \[S(A)_I(R).=\{\phi:A\to R\mid \phi(I)\cdot R =R\}\] + (by the above lemma, these are precisely the points which come from the open subscheme $D(I)$) + \item An \emph{open subfunctor} is a subfunctor $U\hookrightarrow X$ such that for every morphism $S(A)\to X$ the projection map from the (pointwise) fibre product $U\times_X S(A)$ to a subfunctor of the form $S(A)_I$ is an isomorphism for a suitable ideal $I\subseteq A$. + \end{itemize} +\end{definition} + +\begin{bsp} + For $p\neq q$ prime numbers we have $X=\Spec \Z=D(p)\cup D(q)$. But + \[ X(\Z)\supsetneq D(p)(\Z)\cup D(q)(\Z) \,, \] + since both sets on the right are empty. +\end{bsp} + +So this isn't quite right either. Maybe fields? + +\begin{definition} + \begin{itemize} + \item A family $(U_i\hookrightarrow X)_{i\in I}$ of open immersions in $\Psh$ is an \emph{open covering}, if for every field $k$ we have + \[X(k) = \bigcup_{i\in I} U_i(k) \,.\] + (here we could replace "field" by "local ring". The idea is that "points" are specs of fields, and we shouldnt require these covering conditions for all objects, just for points) + \item For ring $A$ a \emph{partition of unity} is given by a finite family $(f_i,x_i)$ with $f_i,x_i\in A$ such that + \[ \sum_i f_i x_i = 1 \,.\] + In this case $(S(A)_{(f_i)})_{i\in I}$ is an open covering of $S(A)$. + \item A presheaf $X\in\Psh$ is \emph{local} (ie. it is a \emph{sheaf}) if for all $A\in \operatorname{CRing}$ and all partitions of unity in $A$ the induced diagram + \[\begin{tikzcd} + X(A) \to \prod_i X(A_{f_i}) \,\substack{\to\\[-1em]\to}\, \prod_{i,j} X(A{f_if_j}) + \end{tikzcd}\] + is a limit-diagram. + \item A \emph{scheme} is a local presheaf that allows an open covering by affine schemes. + \end{itemize} +\end{definition} + +\begin{bem} + An open subfunctor of a scheme is a scheme. +\end{bem} + +\begin{bem} + \begin{itemize} + \item Let $X=(|X|,\OO_X)$ be an locally ringed space. Obtain $S(X)\in\Psh$ with $S(X)(R)=\Hom_{\LRS}(\Spec(R),X)$ + \item For $U\subseteq X$ open $S(U)\subseteq S(X)$ open subfunctor. + \item A covering $X=\bigcup_{i\in I} U_I$ yields an open covering $(S(U_i))_i$ of $S(X)$. + \end{itemize} +\end{bem} + +\begin{satz} + A locally ringed space $X$ is a scheme iff $S(X)$ is a scheme. +\end{satz} + +\begin{bem} + The Vorschrift $X\mapsto S(X)$ defines a functor $\LRS\to\Psh$ that has a left adjoint. +\end{bem} +\end{document}